Symmetric Lie Algebra Research Articles

Geometric realization of the Sasa-Satsuma equation on the symmetric space SU(3)/U(2)

The analytic property of the Sasa-Satsuma equation has been well-explored via using an array of mathematical tools (such as the inverse scattering transformation, the Hirota bilinear method and the Darboux transformation). This paper devotes to exploring geometric properties of this equation via the zero curvature representation in terms of the language in Yang-Mills theory. The generalized Landau-Lifshitz type model of Sym-Pohlmeyer moving curves evolving in the symmetric Lie algebra g=k⊕m with initial data being suitably restricted is gauge equivalent to the Sasa-Satsuma equation. This gives a geometric realization of the Sasa-Satsuma equation.

Deformations of Symmetric Simple Modular Lie (Super)Algebras

We say that a Lie (super)algebra is ''symmetric'' if with every root (with respect to the maximal torus) it has the opposite root of the same multiplicity. Over algebraically closed fields of positive characteristics (up to 7 or 11, enough to formulate a general conjecture), we computed the cohomology corresponding to the infinitesimal deformations of all known simple finite-dimensional symmetric Lie (super)algebras of rank <9, except for superizations of the Lie algebras with ADE root systems, and queerified Lie algebras, considered only partly. The moduli of deformations of any Lie superalgebra constitute a supervariety. Any infinitesimal deformation given by any odd cocycleis integrable. All deformations corresponding to odd cocycles are new. Among new results are classifications of the cocycles describing deforms (results of deformations) of the 29-dimensional Brown algebra in characteristic 3, of Weisfeiler-Kac algebras and orthogonal Lie algebras without Cartan matrix in characteristic 2. Open problems: describe non-isomorphic deforms and equivalence classes of cohomology theories. Appendix: For several modular analogs of complex simple Lie algebras, and simple Lie algebras indigenous to characteristics 3 and 2, we describe the space of cohomology with trivial coefficients. We show that the natural multiplication in this space is very complicated.

Derivations and Central Extensions of Symmetric Modular Lie Algebras and Superalgebras

Over algebraically closed fields of positive characteristic, for simple Lie (super)algebras, and certain Lie (super)algebras close to simple ones, with symmetric root systems (such that for each root, there is minus it of the same multiplicity) and of ranks less than or equal to 8—most needed in an approach to the classification of simple vectorial Lie superalgebras (i.e., Lie superalgebras realized by means of vector fields on a supermanifold),—we list the outer derivations and nontrivial central extensions. When the conjectural answer is clear for the infinite series, it is given for any rank. We also list the outer derivations and nontrivial central extensions of one series of non-symmetric (except when considered in characteristic 2), namely periplectic, Lie superalgebras—the one that preserves the nondegenerate symmetric odd bilinear form, and of the Lie algebras obtained from them by desuperization. We also list the outer derivations and nontrivial central extensions of an analog of the rank 2 exceptional Lie algebra discovered by Shen Guangyu. Several results indigenous to positive characteristic are of particular interest being unlike known theorems for characteristic 0, some results are, moreover, counterintuitive.

Homotopy Rota–Baxter operators and post-Lie algebras

Rota–Baxter operators and the more general \mathcal{O} -operators, together with their interconnected pre-Lie and post-Lie algebras, are important algebraic structures, with Rota–Baxter operators and pre-Lie algebras instrumental in the Connes–Kreimer approach to renormalization of quantum field theory. This paper introduces the notions of a homotopy Rota–Baxter operator and a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra. Their characterization by Maurer–Cartan elements of suitable differential graded Lie algebras is provided. Through the action of a homotopy \mathcal{O} -operator on a symmetric graded Lie algebra, we arrive at the notion of an operator homotopy post-Lie algebra, together with its characterization in terms of Maurer–Cartan elements. A cohomology theory of post-Lie algebras is established, with an application to 2-term skeletal operator homotopy post-Lie algebras.

A Lie algebra representation for efficient 2D shape classification

Riemannian manifold plays a vital role as a powerful mathematical tool in computer vision, with important applications in curved shape analysis and classification. Significant progress has recently been made by Riemannian framework based methods that achieved state-of-the-art classification accuracy and robustness. However, these Riemannian manifold and Lie group methods require a very high computational complexity and do not include a description of the shape regions. This paper presents a novel mathematical tool, called Block Diagonal Symmetric Positive Definite Matrix Lie Algebra (BDSPDMLA) to represent curves, which extends the existing Lie group representations to a compact yet informative Lie algebra representation. The proposed Lie algebra based method addresses the computational bottleneck problem of the Riemannian framework based methods. In addition, it allows the natural fusion of various regions information with curved shape features for a more discriminative shape description. Here the region information is represented by values of distance maps, local binary patterns (LBP) and image intensity. Extensive experiments on five publicly available databases demonstrate that the proposed Lie algebra based method can achieve a speed of over ten thousand times faster than the Riemannian manifold and Lie group based baseline methods, while obtaining comparable accuracies for 2D shape classification.

THE PBW THEOREM FOR AFFINE YANGIANS

We prove that the Yangian associated to an untwisted symmetric affine Kac-Moody Lie algebra is isomorphic to the Drinfeld double of a shuffle algebra. The latter is constructed by the authors in arXiv:1407.7994 as an algebraic formalism of the cohomological Hall algebras. As a consequence, we obtain the Poincare-Birkhoff-Witt (PBW) theorem for this class of affine Yangians. Another independent proof of the PBW theorem is given recently by Guay-Regelskis-Wendlandt.

Drinfeld–Gaitsgory–Vinberg interpolation Grassmannian and geometric Satake equivalence

Let $G$ be a reductive complex algebraic group. We fix a pair of opposite Borel subgroups and consider the corresponding semiinfinite orbits in the affine Grassmannian $Gr_G$. We prove Simon Schieder's conjecture identifying his bialgebra formed by the top compactly supported cohomology of the intersections of opposite semiinfinite orbits with $U(\check{\mathfrak n})$ (the universal enveloping algebra of the positive nilpotent subalgebra of the Langlands dual Lie algebra $\check{\mathfrak g}$). To this end we construct an action of Schieder bialgebra on the geometric Satake fiber functor. We propose a conjectural construction of Schieder bialgebra for an arbitrary symmetric Kac-Moody Lie algebra in terms of Coulomb branch of the corresponding quiver gauge theory.

Factorisation of quasi K-matrices for quantum symmetric pairs

The theory of quantum symmetric pairs provides a universal K-matrix which is an analog of the universal R-matrix for quantum groups. The main ingredient in the construction of the universal K-matrix is a quasi K-matrix which has so far only been constructed recursively. In this paper we restrict to the cases where the underlying Lie algebra is mathfrak {sl}_n or the Satake diagram has no black dots. In these cases we give an explicit formula for the quasi K-matrix as a product of quasi K-matrices for Satake diagrams of rank one. This factorisation depends on the restricted Weyl group of the underlying symmetric Lie algebra in the same way as the factorisation of the quasi R-matrix depends on the Weyl group of the Lie algebra. We conjecture that our formula holds in general.

English

We give a classification of pseudo-Riemannian weakly symmetric manifolds in dimensions 2 and 3, based on the algebraic approach of such spaces through the notion of a pseudo-Riemannian weakly symmetric Lie algebra. We also study the general symmetry of reductive 3-dimensional pseudo-Riemannian weakly symmetric spaces and particularly prove that a 3-dimensional reductive 2-fold symmetric pseudo-Riemannian manifold must be globally symmetric.

Vortex filament on symmetric Lie algebras and generalized bi-Schrödinger flows

In this article, we display an evolving model on symmetric Lie algebras from a purely geometric way by using the Hamiltonian (or para-Hamiltonian) gradient flow of a fourth order functional called generalized bi-Schrodinger flows, which corresponds to the Fukumoto–Moffatt’s model in the theory of moving curves, or the vortex filament in physical words, in $$\mathbb {R}^3$$ . The theory of vortex filament in $$\mathbb {R}^3$$ or $$\mathbb {R}^{2,1}$$ up to the third-order approximation is shown to be generalized to symmetric Lie algebras in a unified way.

A parameterization of the canonical bases of affine modified quantized enveloping algebras

For symmetrizable Kac-Moody Lie algebra $\textbf{g}$, Lusztig introduced the modified quantized enveloping algebra $\dot{\textbf{U}}(\textbf{g})$ and its canonical basis in [12]. In this paper, for finite and affine type symmetric Lie algebra $\textbf{g}$ we define a set which depend only on the root category and prove that there is a bijection between the set and the canonical basis of $\dot{\textbf{U}}(\textbf{g})$, where the root category is the $T^2$-orbit category of the derived category of Dynkin or tame quiver. Our method bases on one theorem of Lin, Xiao and Zhang in [9], which gave the PBW-basis of $\textbf{U}^+(\textbf{g})$.

Upon Generating (2+1)-dimensional Dynamical Systems

Under the framework of the Adler-Gel’fand-Dikii(AGD) scheme, we first propose two Hamiltonian operator pairs over a noncommutative ring so that we construct a new dynamical system in 2+1 dimensions, then we get a generalized special Novikov-Veselov (NV) equation via the Manakov triple. Then with the aid of a special symmetric Lie algebra of a reductive homogeneous group G, we adopt the Tu-Andrushkiw-Huang (TAH) scheme to generate a new integrable (2+1)-dimensional dynamical system and its Hamiltonian structure, which can reduce to the well-known (2+1)-dimensional Davey-Stewartson (DS) hierarchy. Finally, we extend the binormial residue representation (briefly BRR) scheme to the super higher dimensional integrable hierarchies with the help of a super subalgebra of the super Lie algebra sl(2/1), which is also a kind of symmetric Lie algebra of the reductive homogeneous group G. As applications, we obtain a super 2+1 dimensional MKdV hierarchy which can be reduced to a super 2+1 dimensional generalized AKNS equation. Finally, we compare the advantages and the shortcomings for the three schemes to generate integrable dynamical systems.

SHEETS IN SYMMETRIC LIE ALGEBRAS AND SLICE INDUCTION

In this paper, we study sheets of symmetric Lie algebras through their Slodowy slices. In particular, we introduce a notion of slice induction of nilpotent orbits which coincides with the parabolic induction in the Lie algebra case. We also study in more detail the sheets of the non-trivial symmetric Lie algebra of type G2. We characterize their singular loci and provide a nice desingularization lying in so 7.

Integrable systems with unitary invariance from non-stretching geometric curve flows in the Hermitian symmetric space Sp(n)/U(n)

A moving parallel frame method is applied to geometric non-stretching curve flows in the Hermitian symmetric space Sp(n)/U(n) to derive new integrable systems with unitary invariance. These systems consist of a bi-Hamiltonian modified Korteweg-de Vries equation and a Hamiltonian sine-Gordon (SG) equation, involving a scalar variable coupled to a complex vector variable. The Hermitian structure of the symmetric space Sp(n)/U(n) is used in a natural way from the beginning in formulating a complex matrix representation of the tangent space 𝔰𝔭(n)/𝔲(n) and its bracket relations within the symmetric Lie algebra (𝔲(n), 𝔰𝔭(n)).

The noncommutative KdV equation and its para-Kähler structure

We prove that the noncommutative (n × n)-matrix KdV equation is exactly a reduction of the geometric KdV flows from ℝ to the symmetric para-Grassmannian manifold \(\tilde G_{2n,n} \) = SL(2n, ℝ)/SL(n, ℝ) × SL(n, ℝ) and it can also be realized geometrically as a motion of Sym-Pohlmeyer curves in the symmetric Lie algebra sl(2n, ℝ) with initial data being suitably restricted. This gives a para-geometric characterization of the noncommutative matrix KdV equation.

The classification of real forms of simple irreducible pseudo-Hermitian symmetric spaces

The main purpose of this paper is to classify the real forms $M$ of simple irreducible pseudo-Hermitian symmetric spaces $G/R$ with $G$ non-compact. That provides an extension of Jaffee's results (Bull. Amer. Math. Soc. '75; J. Differential Geom. '78), Leung's result (J. Differential Geom. '79) and Takeuchi's result (Tohoku Math. J. '84) concerning the classification of real forms of irreducible Hermitian symmetric spaces of the non-compact type. Moreover, that enables us to classify the pairs of simple para-Hermitian symmetric Lie algebras and their para-holomorphic involutions, which includes Kaneyuki-Kozai's result (Tokyo J. Math. '85) of the classification of simple para-Hermitian symmetric Lie algebras.

Quantum groups via Hall algebras of complexes

We describe quantum enveloping algebras of symmetric Kac-Moody Lie algebras via a nite eld Hall algebra construction involving Z2-graded complexes of quiver representations.

Irregular locus of the commuting variety of reductive symmetric lie algebras and rigid pairs

The aim of this paper is to describe the irregular locus of the commuting variety of a reductive symmetric Lie algebra. More precisely, we want to enlighten a remark of V. L. Popov. In one of his papers, the irregular locus of the commuting variety of any reductive Lie algebra is described and its codimension is computed. This provides a bound for the codimension of the singular locus of this commuting variety. V. L. Popov also suggests that his arguments and methods are suitable for obtaining analogous results in the symmetric setting. We show that some difficulties arise in this case and we obtain some results on the irregular locus of the component of maximal dimension of the “symmetric commuting variety”. As a by-product, we study some pairs of commuting elements specific to the symmetric case, that we call rigid pairs. These pairs allow us to find all symmetric Lie algebras whose commuting variety is reducible.

Unitary similarity to a complex symmetric matrix and its extension to orthogonal symmetric Lie algebras

We present some new characterizations of unitary similarity of a square complex matrix to a symmetric matrix. Our approach uses the singular value decomposition. A result of Vermeer is extended in the context of an orthogonal symmetric Lie algebra of the compact type.

Linearization of systems of four second-order ordinary differential equations

In this paper we provide invariant linearizability criteria for a class of systems of four second-order ordinary differential equations in terms of a set of 30 constraint equations on the coefficients of all derivative terms. The linearization criteria are derived by the analytic continuation of the geometric approach of projection of two-dimensional systems of cubically semi-linear second-order differential equations. Furthermore, the canonical form of such systems is also established. Numerous examples are presented that show how to linearize nonlinear systems to the free particle Newtonian systems with a maximally symmetric Lie algebra relative to \(sl(6, \Re)\) of dimension 35.